Solutions to the Calderón problem exist in a handful of special cases, but the \(C^\infty \) -smooth problem remains open in dimensions greater than two. Belishev [1]} and Lassas–Uhlmann [2]} both show that for surfaces \(S\)  with one boundary component, the classical DN map determines \(S\)  up to conformal class. A better result cannot be achieved since the Laplacian is conformally invariant in dimension 2. Belishev, Badanin, and Korikov recently achieved a similar result for surfaces with multiple boundary components [3]}. It is also known that the DN map for forms recovers partial topological information on arbitrary \(M\)  such as the Betti numbers [4]}, but it is not known whether the DN operator can recover \(M\)  up to homeomorphism. The extension of the DN operator to the complete DN operator in Sharafutdinov and Shonkwiler's paper [5]} is able to recover the absolute and relative cohomologies as well.
